Filtered Legendre expansion method for numerical differentiation at the boundary point with application to blood glucose predictions

Abstract

Let f : [−1, 1] → R be continuously differentiable. We consider the question of approximating f ′ (1) from given data of the form (t j , f (t j )) M j=1 where the points t j are in the interval [−1, 1]. It is well known that the question is ill-posed, and there is very little literature on the subject known to us. We consider a summability operator using Legendre expansions, together with high order quadrature formulas based on the points t j 's to achieve the approximation. We also estimate the effect of noise on our approximation. The error estimates, both with or without noise, improve upon those in the existing literature, and appear to be unimprovable. The results are applied to the problem of short term prediction of blood glucose concentration, yielding better results than other comparable methods.

A Deep Learning Approach to Diabetic Blood Glucose Prediction

“…In this appendix, we review the mathematical background for the method developed in [12] for short term blood glucose prediction. As explained in the text, the main mathematical problem can be summarized as that of estimating the derivative of a function at the end point of an interval, based on measurements of the function in the past.…”

“…Mathematically, if f : [−1, 1] → R is continuously differentiable, we wish to estimate f (1) given the noisy values {y j = f (t j ) + j } at points {t j } d j=1 ⊂ [−1, 1]. We summarize only the method here, and refer the reader to [12] for the detailed proof of the mathematical facts.…”

“…Therefore, it is proposed in [9] to use Legendre polynomials rather than the monomials as the basis for the space of polynomials of degree n. A procedure to choose n is given in [9], together with error bounds in terms of n and the estimates on the noise level in the data, which are optimal up to a constant factor for the method in the sense of the oracle inequality. A substantial improvement on this method was proposed in [12], which we summarize in Appendix A. As far as we are aware, this is the state of the art in this approach in short term blood glucose prediction using linear prediction technology.…”

Filtered spectral differentiation method for numerical differentiation of periodic functions with application to heat flux estimation

“…Numerical differentiation from noisy data is an important issue in a wide range of areas of science and technology. Applications areas include image processing (Wang et al 2002), inverse heat conduction problems (Bozzoli et al 2015(Bozzoli et al , 2017, parameter identification (Hanke Communicated by Antonio José Silva Neto. and Scherzer 1999), viscoelastic mechanics (Sovari and Malinen 2007) and blood glucose predictions (Mhaskar et al 2013). The main difficulty found in practical applications is that numerical differentiation is an ill-posed problem; hence, small input data errors can produce large errors in the computed derivative.…”

Deep Nets for Local Manifold Learning

“…For such problems, the objective is to predict the index f x in the next month, say, based on a vector x of their values over the past few months. Other similar problems include the prediction of blood glucose level f x of a patient based on a vector x of the previous few observed levels [54,51], and the prediction of box office receipts (f x ) on the date of release of a movie in preparation, based on a vector x of the survey results about the movie [57]. It is pointed out in [40,38,19] that all the pattern classification problems can also be viewed fruitfully as problems of function approximation.…”